傳統上，橢球面上的大地線微分方程式常以大地坐標描述之。本研究則利用Euler-lagrange變分法原理推導出以卡式直角坐標所描述的橢球面上大地線微分方程式，並利用Brugnano提出的廣義梯形數值積分法對大地線進行數值積分計算，同時也使用8階的Runge-Kutta演算法與8階的預估改正法來做比較。分析結果顯示：在卡式直角坐標中以廣義梯形數值積分法對大地線進行數值積分運算，不論解算的精準度或效率，都優於另外兩種演算法。

本研究也將廣義梯形數值積分法運用在大地坐標系中，分別對變化大地線長度、起點位置及起始方位角來進行分析。結果顯示：長度20000公里以內的大地線，在兩坐標系裡所計算出的成果差異不大，但隨著大地線長度的增加，可以發現在卡式直角坐標中所算出的成果會逐漸優於在曲線坐標系中的成果。此外在相同的大地線長度下，當起點位置的緯度越高或起始方位角接近0度或180度時，曲線坐標系的計算成果會逐漸變差；而在卡式直角坐標系中，不論大地線的起點位置與起始方位角大小，仍維持相當穏定的計算成果。

Conventionally, the differential equations of geodesic on the rotational ellipsoid are expressed by the geodetic coordinates. In this study, the differential equations are described by Cartesian coordinates and they are derived from the Euler-Lagrange variational principle. We apply the extended trapezoidal rule presented by Brugnano to integrate the differential equations of geodesic and compare the results with Runge-Kutta method and the predictor-corrector method. It is shown from the experiments that the numerical integration by the extended trapezoidal rule is superior to the other methods in precision and efficiency.

The extended trapezoidal rule method is also applied for the differential equations of geodesic described by geodetic coordinates. Three factors are discussed for comparison, namely geodesic length, the starting point position and initial azimuth. The results show that the solutions of both coordinate types are similar when the length of geodesic is within 20000 kilometers and the solutions of Cartesian coordinate type are better than the solutions of geodetic coordinate type if the length of geodesic increases. Besides, the results turn into worse in the geodetic coordinate type, when the initial latitude is higher or initial azimuth near 0 or 180, while in the Cartesian coordinate type, the results of integration are not affected by any initial latitude or initial azimuth.

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